Quantum nature of Wigner function for inflationary tensor perturbations

arXiv:2002.01064v2 [hep-th] 11 Mar 2020

Quantum nature of Wigner function for inflationary tensor perturbations

Jinn-Ouk Gong

and Min-Seok Seo

aKorea Astronomy and Space Science Institute, Daejeon 34055, Korea

bDepartment of Physics Education, Korea National University of Education Cheongju 28173, Korea

Abstract

We study the Wigner function for the inflationary tensor perturbation defined in the real phase space. We compute explicitly the Wigner function including the contributions from the cubic self-interaction Hamiltonian of tensor perturbations. Then we argue that it is no longer an appropriate description for the probability distribution in the sense that quantum nature allows negativity around vanishing phase variables. This comes from the non-Gaussian wavefunction in the mixed state as a result of the non-linear interaction between super- and sub-horizon modes. We also show that this is related to the explicit infrared divergence in the Wigner function, in contrast to the trace of the density matrix.

1 Introduction

Quantum mechanics has been established as a framework for describing nature down to the mi- croscopic scale, yet there still remain many conceptual questions to be answered. In particular, a large gap between the classical and quantum description has brought about the attempts to explain the quantum-to-classical transition dynamically. Remarkably, cosmology is expected to provide a natural testing ground for this issue (for a recent review, see [1]). That comes from the fact that, whereas the primordial fluctuations as we observe from the cosmic microwave background are classical, it is likely to be originated from the quantum fluctuations [2, 3, 4, 5, 6]

as supported by the inflationary cosmology [7, 8, 9].

In order to find out the observable that reveals the quantum origin of the primordial fluc- tuations, we need to make clear the meaning of the quantum-to-classical transition in the cosmological context. Since the typical size of the interaction between gravitons or curvature perturbation is given by H/m_Pl ≪ 1 or even more suppressed by the factor of the slow-roll parameter, the dominant loss of the quantum nature can be discussed in terms of the quadratic action only. More concretely, the quadratic action encodes the incessant interaction between the quantum fluctuations and gravitational background as well. Then a set of quantum fluc- tuations behaves coherently, forming the two-mode squeezed state about which the quantum effects coming from the non-commutativity of field operators are suppressed [10, 11, 12, 13, 14].

Here the two-mode means that the quantum fluctuation with three-momentum k and that with

−k appear pairwise. But still, such a squeezing is a unitary evolution which transforms the pure state into the pure state. Hence we need an additional mechanism that converts the pure state into the mixed state, in which the (diagonalized) density matrix elements correspond to the quantum analog of classical probability distribution. This is achieved by decoherence, in which the system and the environment interact with each other through higher-order non-linear interaction (for reviews, see [15, 16]). In the inflationary universe, the horizon of size 1/H be- comes a natural cutoff distinguishing the system from the environment: an effective field theory below the energy scale H for the super-horizon modes is an open system that interacts with the sub-horizon modes as an environment [17, 18, 19, 20, 21, 22, 23, 24]. Then the evolution of the super-horizon modes is not unitary, as characterized by the Lindblad terms [25]. They describe the transition from the pure to the mixed state in decoherence through the time evolution of the density matrix [26]. One of natural choices of the basis for the density matrix is a set of states obtained from the unitary time evolution of different particle number states. Since the unitary time evolution includes squeezing, each of basis states is the coherent superposition of multiparticle states, but still forms an orthogonal set of basis. Then we have the classical probability that the universe has evolved from different initial states, in which some of quantum fluctuations have excited [24].

With the mechanism described above, one may naively expect that cosmological pertur- bations as we observe are completely described in terms of classical physics. However, unlike the name stands, such a “quantum-to-classical transition” still preserves the quantum nature, from which we may confirm the quantum origin of cosmological perturbations. For example, whereas the density matrix is a quantum analog of the classical probability distribution, they are represented in terms of the quantum states we observe¹. On the other hand, in statistical

1 Since the quantum interference effects disappear through decoherence, one may attempt to explain the collapse of wavefunction after measurement in terms of it. However, the probabilistic nature still remains, so

mechanics, the classical probability distribution is defined in the phase space: it is a func- tion of canonical variables which are eigenvalues of non-commuting quantum operators. The Wigner (distribution) function [27] is devised to construct the classical probability distribution of such type from the density matrix (see [28, 29, 30] for early discussions in the context of cosmology). However, only limited system allows the Wigner function to be interpreted as the probability distribution since its positivity is not guaranteed [31]. The Wigner function is positive definite only when the states comprising the density matrix is represented by the Gaussian wavefunction. In the case of the harmonic oscillator, for example, since the Gaussian wavefunction appears only in the ground state, the Wigner function is positive definite only if we have the pure ground state. Regarding the squeezed state, the squeezing of the vacuum state is represented by the Gaussian wavefunction but when the squeezed states of the initial quantum excitations are mixed, we can find the region in the phase space in which the Wigner function becomes negative. This shows that the probability distribution in the phase space is not always well-defined, reflecting the quantum nature of the primordial fluctuations even after decoherence. Therefore, the negativity of the Wigner function is a signal of quantum origin of the system. We also note that the positivity of the Wigner function does not guarantee the absence of the quantum effect² [32, 33]. For instance, we may find out a Bell inequality violation even if the Wigner function is non-negative. Checking the positivity of the Wigner function is just a basic test for the quantum nature of the system.

In this article, we study the appearance of such negativity of the Wigner function in the context of cosmological perturbations. Our conclusion is that the higher-order non-linear in- teraction gives rise to the possibility for our universe to have the non-Gaussian wavefunction or equivalently, the squeezed states of the initial quantum excitation through the super- and sub-horizon interaction, hence the absolute positivity of the Wigner function is not guaranteed.

Moreover, the negative Wigner function is related to its infrared divergence. This is in contrast to the unit value of the trace of the density matrix, which is the result of the cancellation between the infrared divergence in the no-graviton-excitation state and that in the soft graviton states.

Such an explicit infrared divergence implies that the Wigner function is no longer a meaningful description for the probability distribution of the cosmological perturbations. For this purpose, we consider the tensor perturbations, or the graviton fluctuations, for more quantitative analy- sis. Whereas the cosmological perturbations also contain the curvature perturbation, reflecting a spontaneous breaking of the de Sitter (dS) isometry in quasi-dS spacetime [34, 35, 36, 37], the mechanism for the negativity of the Wigner function is essentially the same as that of the tensor perturbations: the only qualitative difference is a factor of the slow-roll parameter which is an order parameter for the breaking of the dS isometry. In Section 2, we briefly review the squeezing of tensor perturbations, which is useful in our discussions. From this, in Section 3, we present the Wigner function explicitly to show the breaking of its positivity as a result of nonlinear interaction which was intensively studied in [24]. Implication of the Wigner function is visited in Section 4 after which we conclude. We also provide appendix sections to give some details not presented in the main text.

we need to employ another interpretation scheme in addition, such as the many-world interpretation. For more discussion, see, e.g. [16] and references therein.

2We thank referee for pointing out this.

2 Quantum state for tensor perturbations

We consider the spatial metric as

g_ij = a²(τ )(δ_ij + h_ij) , (1)

where dτ = dt/a is the conformal time and hij is the pure tensor perturbations with hⁱi =

∂ihⁱj = 0. Since there are two physical degrees of freedom for hij, we introduce the polarization tensor e_ij(λ), with λ being the polarization index, so that h_ij =P2

λ=1h_λe_ij(λ). For canonical normalization, we introduce

vλ ≡ amPl

√2 hλ, (2)

then there are two copies of the identical action of a canonically normalized scalar field v_λ for each polarization state so from now on we just consider only one polarization state and drop the subscript λ. At quadratic level, adopting the Heisenberg picture, the conjugate pair for tensor perturbations can be written as

ˆ

πk(τ ) = ak(τ )uk+ a^†_−k(τ )u^∗_k, (3) ˆ

vk(τ ) = ak(τ )vk+ a^†_−k(τ )v^∗_k, (4) where the time-dependence is given, not to the mode functions, but to the creation and anni- hilation operators. Here, the initial mode functions u_k and v_k are found to be

u_k= −i rk

2 and v_k = 1

√2k. (5)

What is important here is that we cannot isolate the evolution of the pure sub-system of the mode with k: when we excite the k mode, −k mode is at the same time excited as well. This becomes clear if, from the above expression, we express the creation and annihilation operators in terms of the canonical variables as

ak(τ ) = 1

√2

h√kˆvk(τ ) + i

√kˆπk(τ )i

, (6)

a^†_k(τ ) = 1

√2

h√kˆv−k(τ ) − i

√kπˆ−k(τ )i

, (7)

where it is very important to note that a^†_k(τ ) is given by the canonical pair of not k but −k.

This is obviously because the particle creation always occurs in pair with the opposite momenta.

To isolate the mode with k from that with −k, we “define” the position ˜q^k and momentum ˜pk

which are purely written in terms of the operators with the corresponding momentum k, not involving the opposite momentum −k as

˜

qk ≡ 1

√2k

hak(τ ) + a^†_k(τ )i

, (8)

˜

pk ≡ −i rk

2

hak(τ ) − a^†k(τ )i

. (9)

Now we make two steps further. First, we “discretize” the momentum as R

d³k/(2π)³ → L⁻³P

k where L³ is the volume under consideration. With a given volume L³, we may now isolate the volume from the canonical creation and annihilation operators ak and a^†_k in such a way that, since ak has a mass dimension of −3/2, we define

ak ≡ L^3/2ˆak (10)

and likewise for a^†_k. That means, the new dimensionless operators ˆak and ˆa^†_k now satisfy the following commutation relation: h

ˆak, ˆa^†_k^′i

= δkk^′. (11)

Second, we note that ˜qk and ˜pk are of mass dimension −2 and −1 respectively. To apply directly the wisdom of the standard quantum harmonic oscillators, we rescale the dimensionful factors to introduce the dimensionless position ˆqk and momentum ˆpk as

ˆ qk ≡

r k

L³q˜k= 1

√2

hˆak(τ ) + ˆa^†_k(τ )i

, (12)

ˆ

pk ≡ 1

√kL³p˜k= − i

√2

hˆak(τ ) − ˆa^†k(τ )i

. (13)

This new pair is obviously Hermitian, i.e. ˆq_k^† = ˆqk and ˆp^†_k = ˆpk, and satisfies the canonical commutation relation: 

ˆ qk, ˆpq

 = i

ˆak, ˆa^†_q

= iδkq. (14)

We note that typical cosmological observables are correlators of field operators with respect to the initial vacuum, from which we can obtain information on (12) and (13) indirectly. However, the Hermiticity of these operators implies that there must in principle be a clever way to

“measure” them.

Given the canonical variables (12) and (13), we can now make use of the position basis vector |q^k, q−ki for the two-mode state with k and −k. From noting that bH =P

kHb^k with Hb^k = 1

2

 k

1 + ˆa^†_kˆak+ ˆa^†_−kaˆ−k

+ ia^′ a

− ˆa^kˆa−k+ ˆa^†_−kˆa^†_k

, (15)

we can write the (free) evolution operator as a product of bUk which arises solely from bH^k: Ub0(τ, τ0) = exp



−i Z τ

τ0

Hb0(τ^′)dτ^′



=Y

i

Ubkⁱ(τ, τ0) . (16)

Then the state |Ψi that has evolved from the initial vacuum |0i at τ0,

|Ψi = bU0(τ, τ0)|0i = Y

i

Ubki(τ, τ0)|0i , (17)

using the complete basis |q^k, q−ki, can be written

|Ψi =Y

i

Z

dqkidq−ki

q^ki, q−ki

EDqki, q−ki

 bUki(τ, τ0)0E

=Y

i

Z

dqkidq−kiΨ(qki, q−ki)|q^ki, q−kii , (18)

where we have defined the wavefunction Ψ(qk, q_−k) in the position basis |q^k, q_−ki as the rep- resentation of |Ψi. The evolution operator bUk(τ, τ0) can be written as the product of the

“rotation” operator bRk and the “squeezing” operator bSk, i.e. bUk = bSkRbk. Further, while the only role of bRk is to change the phase, bSk is as the name stands solely responsible for the two-mode squeezing of the initial vacuum state. More concretely, in the rotation operator

Rbk= exph

−iθ^k

1 + ˆa^†_kˆak+ ˆa^†_−kˆa_−ki

, (19)

the exponents ˆa^†_kˆak and ˆa^†_−kaˆ−k are the number operators counting the number of k and −k mode excitations, respectively. Then when bRkis applied to |0i, it just provides an overall phase e^−iθk. In the density matrix we will obtain in Section 3, the phase does not play any role in the matrix element for bUk|0ih0| bU_k^†= e^−iθkSbk|0ih0| bS_k^†e^+iθk = bSk|0ih0| bS_k^†. In the same way, the phase does not contribute to the diagonal density matrix elements like bUkˆa^†_k|0ih0|ˆa^kUb_k^† as well. Thus, from now on we drop the irrelevant rotation operator and consider only bSk in the wavefunction.

The phase will be written explicitly when the off-diagonal density matrix element is discussed [see (44)]. The wavefunction we are interested in is given by

Ψ(qk, q−k) =D qk, q−k

 bSk(τ, τ0)0E

=

* qk, q−k

1 cosh(rk/2)

X

n



− e^2iϕktanh

rk

2

nn, k; n, −k +

, (20)

where the n-particle excited state, with the momenta k and −k each, is

|n, k; n, −ki ≡ 1 n!

aˆ^†_−kˆa^†_kn

|0i . (21)

Then it can be shown that Ψ(qk, q−k) is given by

Ψ(qk, q−k) = e^{A(rk,ϕk)(q2k+q−k2 )−B(rk,ϕk)qkq−k} cosh(rk/2)√

π q

1 − e^4iϕktanh²(rk/2)

, (22)

A(rk, ϕk) = e^4iϕktanh²(rk/2) + 1 2

e^4iϕktanh²(r_k/2) − 1 , (23) B(r_k, ϕ_k) = −2e^2iϕktanh(rk/2)

e^4iϕktanh²(rk/2) − 1. (24)

Here, r_k and ϕ_k are the parameters of the Bogoliubov transformation given by (B.4) and (B.5).

The detail of the derivation of (22) is given in [38] (see also [39]).

3 Wigner function for tensor perturbations

Now we consider explicitly the reduced density matrix ρred of the inflationary tensor perturba- tions on super-horizon scales under the influence of cubic interaction with sub-horizon modes.

We obtain the Wigner function from ρred. If we ignore cubic or higher non-linear interactions,

super- and sub-horizon modes behave independently, then the two-mode squeezed state com- prised of super-horizon modes form the pure state. Since the wavefunction for the squeezed vacuum state in (22) is Gaussian, the Wigner function for this state is also Gaussian and pos- itive definite. In this case, it has been shown that the two-point correlators in terms of the quantum and the classical stochastic formalism are not distinguishable regardless of squeezing [30]. On the other hand, studies in this section show that when we take into account the non-linear interaction, the inevitable interaction between super- and sub-horizon modes results in the violation of the positivity of the Wigner function. This indicates that the Wigner func- tion cannot be used as a conceivable probability distribution under decoherence, invalidating correlators defined upon the Wigner function as physical quantities.

Under the interaction between the super- (“system” S) and sub-horizon modes (“environ- ment” E), tracing out the environmental sub-horizon modes introduces the non-unitary terms in the Lindblad equation:

d

dτρred(τ ) ⊃ −1 2

X

i

hL^†₁L2ρred(τ ) + ρredL^†₂L1− 2L¹ρredL^†₂ + (L1 ↔ L²)i

, (25)

where dots indicate the irrelevant unitary evolution terms and the Lindblad operators L1 and L2 are defined in terms of the free, unitary evolution operator for the environment bU0,E with the subscript 0 standing for the free evolution and the interaction Hamiltonian Hint by [22, 24]

L1 ≡D

EⁱH^int,SUb0,E(τ ; τ0)E(τ⁰)E , L2 ≡

 Eⁱ

Z τ τ0

dτ1Hint,I(τ1 − τ) bU0,E(τ ; τ0)  E(τ⁰)

 .

(26)

Here the interaction Hamiltonian with subscripts S and I correspond to those written in the Schr¨odinger and interaction picture, respectively. Since the interaction Hamiltonian is dom- inated by cubic term for H/m_Pl ≪ 1, L^†1L₂ or L^†₂L₁ in (25) provide at most six creation operators for super-horizon modes depending on how many sub-horizon excitations are traced out in (26). Since these excitations act on the initial vacuum, they evolve in time by the free unitary operator bU0,S dominantly. From now on, we omit the subscript S as obviously we only consider the evolution of super-horizon modes. Hence, to the lowest order in H/mPl, ρred for each super-horizon mode k is written in the schematic form

ρred = X

m,n≤6

ρmnUb0a^†₁· · · a^†m|0ih0|a^1′· · · aⁿUb₀^†, (27) where the evolution operator, creation and annihilation operators and the vacuum state are those for the super-horizon modes. Thus we can see that ρred contains inherently a set of basis

|ai =n

Ub0|0i, bU0a^†_1,|0i, bU0a^†₁a^†₂|0i, · · ·o

, (28)

and using this basis can be written in the matrix form, with the subscript ab denoting the element at a-th row and b-th column in the matrix,

ρred|^ab = ha|ρ^red|bi =







1 − ρ00 0 ρ₀₂

0 ρ11 0 03×4

ρ20 0 0

04×3 04×4





 , (29)

where ρ₂₀= ρ^∗₀₂. The result is written to the leading order, O(H²/m²_Pl). If we take the quartic or higher interactions into account, the components for the reduced density matrix will be filled more. For the detailed calculation, see [24]. Note that only for the 00 component the evolution operator acts upon the vacuum state, it is the only one for which we can apply directly the wavefunction (22) which, as can be seen from (20), has evolved from the vacuum state. Other components such as 11 are from the excited initial states, so require more care.

00 component

First we consider the simplest case – the 00 component:

ρ_red|00= 1 − ρ00
bU₀|0ih0| bU₀^†, (30) where the coefficient is given by [24]

ρ00 = − 36

(2π)³δ⁽³⁾(q)H² m²_Pl

4 9τ³8π²



0.577148 − 32 45log ε



, (31)

with the infrared cutoff in the Planck unit ε ≪ 1.

Since we are dealing with a two-mode excited state, we need to expand the simplest Wigner function (A.1) to include the other, opposite momentum −k as [1]

W (q1, q2; p1, p2) = Z

dxdye^−ip1xe^−ip2y

 q1+ x

2, q2+ y 2  ρ

 q¹−x

2, q2− y 2

. (32)

As we have noted, concentrating only on bSk for the evolution operator with the subscript 1 and 2 denoting respectively k and −k, the contribution of ρ^red|⁰⁰ to the Wigner function becomes W₀₀(qk, q_−k; pk, p_−k) = 1 − ρ00

Z dxdye^−ipkxe^−ip−kyΨ

 qk+ x

2, q_−k+y 2

 Ψ^∗

 qk−x

2, q_−k− y 2

 . (33) Note that as mentioned before we can write the Wigner function in terms of the wavefunction only for the 00 component. After some calculations, with the detail given in Appendix C, we find

W00(qk, q−k; pk, p−k) = 4 1 − ρ⁰⁰
expn

− p²k+ p²_−k+ q²_k+ q²_−k

cosh rk

+ 2h

pkp_−k− q^kq_−k

cos(2ϕ_k) − 2 p^kq_−k+ p_−kqk

sin(2ϕ_k)i

sinh r_ko

≡ 4 1 − ρ00

wk. (34)

The exponent for the exponential is a four-variable function (pk, p−k, qk, q−k) and is thus hard to visualize. But nevertheless what is easy to note is that the whole exponential factor is always positive definite. Thus we conclude that the Wigner function for the 00 component (34) is always positive.

11 component

Now we move to the first non-trivial part, the 11 component of ρred. Schematically, from (27) ρred|¹¹ = ρ11Ub0a^†_qa|0ih0|a^qbUb₀^†, (35) where the coefficient is given by [24] (with typo corrected)

ρ₁₁= −18δ⁽³⁾(q_ab)δ_λaλb

H² m²_Pl

4 9τ³

2π

q³ × 16 525¯q³



8¯q⁵− 70¯q + 35 log

1 + ¯q 1 − ¯q



, (36)

where qa = −q^b ≡ q and ¯q ≡ q/H < 1. Before we proceed the practical calculations, let us pause and see what this element should mean. From (16), we can see that only the Hamiltonian density with the momentum qa survives, as [see (C.9) for the operator dependence of bU0]

Ub0a^†_q|0i ∼

Z d³k

(2π)³ a^†_kak

a^†_q|0i =

Z d³k (2π)³a^†_k

ak, a^†_q

+ a^†_qak

|0i = a^†q|0i . (37)

Thus among all k-modes contained in the Hamiltonian, only the mode which has the same momentum as the external one survives. This can be more directly understood by noting that in (35), the evolution operator bU0 is operational on the state

a^†_q|0i ≡ |1^qi , (38)

the one-particle excited state with the momentum q that belongs to the system sector, q < H.

As the evolution operator is a free one, only the mode with q is responding to |1^qi and all the other modes simply disappear. From now on, without losing generality we identify the super-horizon external momentum q as k.

Now let us proceed the practical calculations for the Wigner function for the 11 component.

Rewriting (35) gives

ρred|¹¹= ρ11Sbka^†_k|0ih0|a^kSb_k^†, (39) where we have dropped the rotation operator bRk in the evolution operator bUk = bRkSbk. Here, for the annihilation operator it should be a−k due to the delta function in ρ11 as shown in (36), but as the squeezing operator bSkwill excite both k and −k modes, we do not distinguish k and

−k assigned to the operators. Since bSk is operating on not |0i but a^†k|0i, we cannot directly apply the wavefunction (20) as we did for the 00 component. Instead, we have to calculate how bSk evolves the one-particle excited state a^†_k|0i. The detail of the calculations is given in Appendix C, and we find

W11(qk, q−k; pk, p−k) = 4ρ11wk

n− 1 + p²k− p²−k+ q²_k− q²−k+ p²_k+ p²_−k+ q²_k+ q²_−k

cosh rk

+ 2h

(pkq−k+ p−kqk) sin(2ϕk) + (qkq−k− p^kp−k) cos(2ϕk)i

sinh rk

o. (40) This is a complicated four-variable function of (pk, p−k, qk, q−k) so not easy to visualize. But nevertheless we can collect the relevant parts. For this purpose, we note that the exponential factor is the same as W00 and is always positive definite. Thus, the only part we need to see if negative at all is the polynomial terms inside the curly brackets. They are, for any moderate value of the squeezing parameter rk, exponentially positive. Nevertheless, they become negative for very small values of (pk, p−k, qk, q−k) all around zero. We will return to this issue in Section 4.

20 component

Now we move to the 20 component of ρred. Schematically, from (27) it is

ρ_red|20= ρ₂₀Sbka^†_kaa^†_kb|0ih0| bS_k^†, (41) where the coefficient is given by

ρ20= ρ11e^2ikτ. (42)

Again, we have dropped the rotation operator bRk in the evolution operator bUk = bRkSbk. As before, since bSk is operating on not |0i but

1

2a^†_kaa^†_kb|0i = 1

2a^†_−ka^†_k|0i ≡ |2^ki , (43) we cannot directly apply the wavefunction (20) as we did for the 00 component. Instead, we have to calculate how bSk evolves the two-particle excited state a^†_−ka^†_k|0i. The detail is given in Appendix C, and we find W20 as

W20(qk, q−k; pk, p−k)

= 2ρ20wk

"

2 cos(2ϕk) sinh rk p²_k+ p²_−k+ q_k² + q_−k²

+ 4 cos(2ϕk) sin(2ϕk)(cosh rk− 1) p^−kqk+ pkq−k

+ 4h

sin²(2ϕk) + cos²(2ϕk) cosh rk

i qkq−k− p^kp−k


− i



2 sin(2ϕ_k) sinh r_k p²_k+ p²_−k+ q_k²+ q_−k²
+ 4h

cos²(2ϕ_k) + sin²(2ϕ_k) cosh r_ki

pkq_−k+ p_−kqk

+ 4 sin(2ϕk) cos(2ϕk) cosh rk− 1

qkq−k− p^kp−k

#

e^−3iθk, (44)

where the factor e^−3iθk at the end comes from the rotation operator (19) acting on |2^ki. Note that with the coefficient of the 02 component of ρred being given by ρ02= ρ^∗₂₀, computing W02

explicitly gives

W02(qk, q−k; pk, p−k) = W₂₀^∗(qk, q−k; pk, p−k) . (45) Total Wigner function

Now we consider the total Wigner function for the reduced density matrix. To begin with, we note that the Wigner function elements (34), (40) and (44) are all for a single mode with k.

The “total” Wigner function of the system of our interest should include all the modes. To see how the contributions from other momentum modes are included, let us consider simply two modes, k1 and k2:

ρ = ρ0000|0¹02ih0¹02| + ρ¹⁰⁰⁰|1¹02ih0¹02| + ρ¹⁰¹⁰|1¹02ih1¹02| + · · · , (46) where |nⁱi corresponds to the n-particle excited state with the momentum kⁱ, e.g. |1¹i = |1^k1i, and ρijklis an arbitrary coefficient. Now, to compute the “total” Wigner function of this system,

since there are two momenta³, we introduce two dummy integration variables s₁ and s₂ such that

W = Z

ds1ds2e^−ip1s1e^−ip2s2

q1+s1

2, q2+ s2

2  ρ

q¹− s1

2, q2− s2

2

= Z

ds1e^−ip1s1

ρ₀₀₀₀ 2

q1+ s₁ 2  0¹

01

q¹− s₁ 2

+ ρ1000

q1 +s₁ 2  1¹

01

q¹−s₁ 2

+ ρ1010

q1+s₁ 2  1¹

11

q¹−s₁ 2

 + · · ·

 Z

ds2e^−ip2s2

q2+s₂ 2  0²

02

q²−s₂ 2

+ (k1 ↔ k²) , (47)

where we have written in such a way that the Wigner function of each mode is more manifest.

We should first note that when we are exciting a certain mode, the others are in their vacuum states. Thus the terms inside the square brackets, multiplied by the vacuum-vacuum component of the Wigner function for the k₂ mode is the total Wigner function only for the k₁mode, except that the component for which all the modes are in the vacuum states are equally distributed, in the current case divided by two as there are two modes. This is because, as the case for which all the modes are in the vacuum is unique, this should equally contribute to the Wigner function of each mode. This suggests that for a huge number of discrete modes we should divide (34) by the volume of the super-horizon momentum space with the ultraviolet cutoff given by H, V_SH = 4πH³/3 × 1/2, with the factor 1/2 being due to the pairwise appearance of k and

−k in the two-mode squeezed state.

Then, (47) is written as W =

1

2W00(qk1, q−k¹; pk1, p−k¹) + W11(qk1, q−k¹; pk1, p−k¹) + · · ·



w2+ (k1 ↔ k²) . (48)

3Precisely speaking, there are two excitations each, with positive and negative momenta. Further, as there are infinitely many discrete momenta, the whole state can be written as

(n^k₁, n_−k1); (n^k₂, n_−k2); · · ·

= ⊗_i=1

nki, n_−ki

,

so that the states given in (46) [as well as in (38) and (43)] are in fact

|0ⁱi =

0^ki,0−ki

, |1ⁱi = a^†ki

0^ki,0−ki

=

1^ki,0−ki

, |2ⁱi = 1

2a^†_−kia^†_ki

0^ki,0−ki

=

1^ki,1−ki

, and so on.

Expanding ρ₂₀ = ρ₁₁e^2iqτ, we may just sum W₁₁, W₂₀ and W₀₂ to find W11+ W20+ W02 = W11+ 2ℜ W²⁰

= 4ρ11wk



− 1 + p²k− p²−k+ q²_k− q−k² + p²_k+ p²_−k+ q_k² + q_−k²

cosh rk

+ 2h

(pkq_−k+ p_−kqk) sin(2ϕ_k) + (qkq_−k− p^kp_−k) cos(2ϕ_k)i

sinh r_k + cos(2kτ − 3θ^k)n

2 cos(2ϕ_k) sinh r_k p²_k+ p²_−k+ q_k² + q_−k²

+ 4 cos(2ϕ_k) sin(2ϕ_k)(cosh r_k− 1) p−kqk+ pkq_−k
+ 4h

sin²(2ϕk) + cos²(2ϕk) cosh rk

i qkq−k− p^kp−k

o

+ sin(2kτ − 3θ^k)n

2 sin(2ϕk) sinh rk p²_k+ p²_−k+ q_k² + q_−k²
+ 4h

cos²(2ϕk) + sin²(2ϕk) cosh rk

i pkq−k+ p−kqk

+ 4 sin(2ϕk) cos(2ϕk) cosh rk− 1

qkq−k− p^kp−k

o . (49) Thus, (48) can be written as

W = 4

1 − ρ⁰⁰ 2 + ρ11

h− 1 + p²k1 − p²−k¹ + q_k²1 − q²−k¹ + · · ·i

w1w2+ (k1 ↔ k²) . (50) This can be immediately extended to a large number of discrete momenta to give

W = 4 1−ρ00

w₁w₂w₃· · ·+4X

i

ρ₁₁(k_i)h

−1+p²ki−p²−ki+q²_ki−q−k² i+· · ·i

w₁w₂w₃· · ·+all perms . (51) Now using the de Sitter solutions for rk and ϕk given respectively by (B.4) and (B.5), we find

sinh r_k = 1

2kτ , cosh r_k=

r

1 + 1 4k²τ² , sin(2ϕk) = 2

r

4 + 1 k²τ²

, cos(2ϕk) = 1 kτ

r

4 + 1 k²τ²

, (52)

then it is trivial to find the explicit time dependence. Especially, in the late-time limit kτ → 0, we find

W₁₁+ W₂₀+ W₀₂

|kτ |≪1−→ 4ρ11exp

 1 2kτ

h(pk− p^−k)²+ (qk+ q−k)²i

+ 4(pkq−k+ p−kqk) + O(kτ)



×



− 3 2kτ

h(pk− p^−k)²+ (qk+ q−k)²i

− 1 − 6(p^kq−k+ p−kqk) + p²_k− p²−k+ q²_k− q−k² + O(kτ)

 . (53)

This shows that for non-zero value of (qk, q_−k, pk, p_−k), W₁₁ + W₂₀+ W₀₂ becomes large and positive at late time, −kτ → 0, as the first term in the curly bracket is dominant. On the other hand, as all the value of (qk, q−k, pk, p−k) approach zero, only the second term, −1 in the curly bracket remains, which results in the negative Wigner function.

4 Discussions

In the previous section, we have computed the Wigner function contributions from the non- linearly evolved elements in the reduced density matrix of tensor perturbations. While the exponential factor, which the 00 component also contains, remains positive definite, the factor

−1 in the polynomial terms in (49) can make the Wigner function negative for small enough values of (pk, p−k, qk, q−k). One may then argue naively as follows. These variables are written essentially in terms of the mode function and its time derivative. In any reasonable model of inflation, typically the mode function becomes frozen on super-horizon scales and accordingly its time derivative almost vanishes. Further, the mode function amplitude contains the factor H/mPl ≪ 1. Thus, very naively, it seems that we are naturally led to have vanishingly small (pk, p−k, qk, q−k). Interestingly, Wigner function becomes negative around this region, which signals the quantum nature of the cosmological perturbation still remains even after the horizon excape.

Before getting into detail, we make one point clear. For this purpose, let us consider the Bogoliubov transformation (B.1), where the time- and momentum-dependent coefficients α_k(τ ) and βk(τ ) are subject to the constraint (B.2). The well-known “mode function” vk(τ ) is given by the coefficient of the initial creation and annihilation operators: from (4) and (B.1), we can write

ˆ

vk= ak(τ )vk+ a^†_−k(τ )v_k^∗ = 1

√2k

αk(τ ) + β_k^∗(τ )

| {z }

=(¹⁻kτⁱ )^e−ikτ

ak(τ0) + 1

√2k

αk(τ ) + β_k^∗(τ )∗

a^†_−k(τ0)

= ak(τ₀)v_k(τ ) + a^†_−k(τ₀)v^∗_k(τ ) . (54)

Thus, while both αk(τ ) and βk(τ ) contain the plane-wave e^±ikτ, the mode function vk(τ ) as we know is given by a specific combination of them. Meanwhile, from (12) and (13)

ak(τ ) + a^†_k(τ ) = αk(τ )ak(τ0) + βk(τ )a^†_−k(τ0) + c.c. , (55) ak(τ ) − a^†k(τ ) = αk(τ )ak(τ0) + βk(τ )a^†_−k(τ0) − c.c. , (56) we can read that 1) the position and momentum operators ˆqk and ˆpq have no specific combi- nation of αk(τ ) and βk(τ ) that gives the mode function solution vk(τ ) and thus they cannot be written in terms of vk(τ ), and 2) ˆqk and ˆpq contain both initial annihilation and creation operators for both k and −k modes, and thus are superpositions of ±k modes.

Then what about the state |q^k, q−ki? As in the case of textbook quantum harmonic oscil- lator, the state upon which the annihilation and creation opeartors act is the particle number state: the state that denotes 0 particle with a specific momentum k (ak|0i = 0), one-particle (a^†_k|0i = |1^ki), and so on. Thus the position state |q^ki and the particle number state |n^ki – the collection of states that can be written in terms of the vacuum |0i and the creation operator

a^†_k – are independent. What we can do is to write one state in terms of the other as, given the competeness P

n|n^kihn^k| = 1,

|q^ki =X

n

|n^kihn^k|q^ki =X

n

Ψ^∗_n(qk)|n^ki , (57) where Ψ_n(qk) is the “wavefunction”, such as (A.12) for one-dimensional harmonic oscillator.

Now, we can address what is the nature of qk and so on that appear in (49). Inserting (21)

˜

qk|q^k, q−ki = 1

√2k

hαk(τ )ak(τ0) + βk(τ )a^†_−k(τ0) + c.c.iX^∞

n=0

|n, k; n, −ki hn, k; n, −k|q| {z ^k, q−k}i

=Ψ^∗n(qk,q−k)

= infinite sum of |n − 1; ni, |n; n + 1i, |n + 1; ni, |n; n − 1i . (58) Thus we conclude we find a collection of infinitely many excitations, and unlike the naive expectation we have seen at the beginning of this section, no direct relation to the mode function vk(τ ) can be found.

Since the wavefunction of the squeezed vacuum (22) is Gaussian while those of the squeezed excitations are not [see (C.25) and (C.35)], the Wigner function is not positive definite. Even worse, the Wigner function is not free of the infrared divergence. These show that Wigner function as the classical probability distribution in the phase space is not well-defined contrary to the density matrix, which is positive definite and infrared finite. To be more quantitative on the infrared finiteness, consider the Wigner function at qk = q−k = 0 and pk = p−k = 0, then we have

W (0, 0; 0, 0) = Z

dxdyDx 2,y

2  ρ − x

2, −y 2

E. (59)

Comparing this to the trace of the density matrix which is taken over the (qk, q−k) space [we denote them by (x, y) for comparison],

Tr[ρ] = 1 4

Z

dxdyDx 2,y

2  ρ

 x

2,y 2

E, (60)

we find that the crucial difference is the additional negative sign in the ket state. For the density matrix to be well-defined description for the mixed state probability distribution, the trace of the density matrix is infrared finite by the cancellation between the infrared divergent parts of ρ00 and ρii with i ≥ 1. This comes from the fact that the extremely soft excitations are not distinguishable to the vacuum state, which is originally used to argue the infrared finiteness of the quantum transition amplitude [40]. Then we can normalize the trace by Tr[ρ] = 1 reflecting the total probability is given by 1. Explicit calculation in [24] shows that at least to the order of H²/m²_Pl, the infrared finiteness of the trace,

Tr[ρ] = 1 − ρ⁰⁰
1 4

Z

dxdyDx 2,y

2  0ED

0x 2,y

2 E+

Z d³q (2π)³ρ11

1 4

Z

dxdyDx 2,y

2  1^qED

1q

 x

2,y 2

E

+

Z d³q (2π)³ρ20

1 4

Z

dxdyDx 2,y

2 

2^q,−qED 0x

2,y 2

E+ c.c.

= 1 − ρ⁰⁰
+

Z d³q

(2π)³ρ11, (61)

where R

d³q/(2π)³ρ₁₁(q) = ρ₀₀, is well satisfied from the cancellation between infrared diver- gences in ρ00 and ρ11. Moving to W (0, 0; 0, 0), since the first (second) excitation state is odd (even) under x → −x and y → −y, i.e. h−x, −y|1i = −hx, y|1i and h−x, −y|2i = hx, y|2i, we find easily

W (0, 0; 0, 0) = 4 1 − ρ⁰⁰

− 4

Z d³q

(2π)³ρ11 = 4 1 − 2ρ⁰⁰

. (62)

This can be checked explicitly from (34), (40) and (44): putting qk = q−k = 0 and pk= p−k= 0, we obtain respectively W00(0, 0; 0, 0) = 4 1 − ρ⁰⁰

, W11(0, 0; 0, 0) = −4ρ¹¹, and W20(0, 0; 0, 0) = 0, which is consistent with the above. The relative minus sign between two terms shows that the infrared divergence in ρ11 is no longer cancelled with that in ρ00 hence W (0, 0; 0, 0) cannot be infrared finite.

We can also see that if ρ00 > 1/2, W (0, 0; 0, 0) becomes negative. From (34), this is possible either for large value of H²/m²_Plor small value of ε. For large H²/m²_Pl, however, we should expect the next-to-leading corrections in terms of the interaction Hamiltonian of O(H⁴/m⁴_Pl) should be significant so that our perturbative results should be modified. Meanwhile, the term containing the infrared cutoff ε may call for different perturbative expansion in terms of (H²/m²_Pl) log ε and even resummed to some analytic function as can be seen in QCD. No matter how ε behaves, the point is that the infrared cutoff ε is introduced to regulate the infrared divergence, which is eventually cancelled by the soft graviton contribution in ρ11 to make Tr[ρ] = 1 while not in the Wigner function. This indicates that the standard interpretation of the Wigner function as the classical probability in the phase space is invalid because of not only being negative, but also exhibiting “naked” infrared divergence.

5 Conclusions

In this article, we have tested whether the Wigner function can be a conceivable physical quantity measuring the probability distribution on the real phase space of the primordial tensor perturbation. We first observe that even our universe has started from the Bunch-Davies vacuum, it would eventually become the mixed state containing the squeezed states of the initial excitations as a result of the non-linear interaction. The wavefunctions of squeezed states are not Gaussian unless the Bunch-Davies vacuum is squeezed, indicating that the positivity of the Wigner function is not guaranteed. We find that around the vanishing values of the real phase space variables (qk, q−k, pk, p−k), the negativity of the Wigner function is evident. This in fact has to do with the cancellation of the infrared divergence to achieve Tr[ρ] = 1. Such a cancellation does not take place in the Wigner function, which not only leads the Wigner function to the negative value, but also makes it uncontrollably divergent. These two enable us to conclude that the Wigner function is not a good description to the probability distribution of the cosmological perturbations, indicating that the quantum nature is not completely lost even after decoherence.

Acknowledgements

We thank Sugumi Kanno, J´erˆome Martin and Jiro Soda for discussions while this work was un- der progress. JG is supported in part by the Mid-career Research Program (2019R1A2C2085023)

through the National Research Foundation of Korea Research Grants. JG also acknowledges the Korea-Japan Basic Scientific Cooperation Program supported by the National Research Foun- dation of Korea and the Japan Society for the Promotion of Science (2018K2A9A2A08000127), and the Asia Pacific Center for Theoretical Physics for Focus Research Program “The origin and evolution of the Universe” where parts of this work were presented and discussed.

A Wigner function and harmonic oscillator

In this section, we recall the basic of the Wigner function. Given a pair of canonical variables (q, p), the Wigner function W (q, p) is defined by

W (q, p) ≡ Z ∞

−∞

dse^−ips

 q + s

2  ρ

 q − s

2

, (A.1)

where ρ is the density matrix. Equivalently, by noting that ρ ≡ |ΨihΨ| with |Ψi being the state of the system, and that the representation of the state |Ψi in the position space x is given by Ψ(x) = hx|Ψi,

W (q, p) = Z ∞

−∞

dse^−ips

 q + s

2  Ψ

Ψ  q −

s 2

= Z ∞

−∞

dse^−ipsΨ

 q + s

2

 Ψ^∗

 q − s

2



. (A.2)

Then, clearly W (q, p) is real. If integrated over either position or momentum, we find Z dq

2πW (q, p) = hp|ρ|pi , (A.3)

Z dp

2πW (q, p) = hq|ρ|qi . (A.4)

Given a Hermitian operator A, we define the so-called “Weyl transformation” A(q, p) as, similar to the Wigner function,

A(q, p) ≡ Z

dse^−ips

 q + s

2  A

 q − s

2

. (A.5)

Integrating the product of A and the Wigner function W over q and p gives Z dqdp

2π A(q, p)W (q, p) = Z

dqhq|Aρ|qi = Tr(Aρ) . (A.6)

We now consider the quantum harmonic oscillator with m = ω = 1. The Hamiltonian operator is written as

H =b pˆ² 2 + qˆ²

2 , (A.7)

where the pair (ˆq, ˆp) is canonical:  ˆ q, ˆp

= i . (A.8)

We can define the annihilation and creation operators respectively by ˆa ≡ q + iˆˆ p

√2 , (A.9)

ˆa^†≡ q − iˆpˆ√

2 , (A.10)

which satisfy the canonical commutation relation:

ˆa, ˆa^†

= −i ˆ q, ˆp

= 1 . (A.11)

Then the normalized wavefunction in the position space Ψ(x) = hx|Ψi is found to be Ψn(x) = 1

√2ⁿn!

e^−x2/2

π^1/4 Hn(x) , (A.12)

where n is an integer and Hn(x) is the Hermite polynomial.

With the wavefunction Ψ_n(x), we can compute straightly the Wigner function for each state.

In the table below we show the results for n = 0, 1 and 2 explicitly:

n Ψ_n(x) = hx|Ψ = ni W_n(q, p) = Z ∞

−∞

dse^−ipsΨ_n q + s

2

Ψ^∗_n q − s

2

 Region for W_n(q, p) ≤ 0

0 e^−x2/2

π^1/4 2e^−(q2+p2) none

1

√2

π^1/4xe^−x2/2 h

4 q²+ p²

− 2i

e^−(q2+p2) q² + p² ≤ 1 2

2 1

√2π^1/4 2x²− 1

e^−x2/2 2h

2 q²+ p²
2

− 4 q²+ p²
+ 1i

e^−(q2+p2) 1 − 1

√2 ≤ q²+ p² ≤ 1 + 1

√2

As we consider higher excited states, the corresponding Wigner functions become more com- plicated. But they all exhibit negative values in certain regime, not always positive definite.

B Bogoliubov transformation

The general solutions for the Hamiltonian equations for the creation and annihilation operators are given by the linear combination of the initial ones:

ak(τ ) = α_k(τ )ak(τ₀) + β_k(τ )a^†_−k(τ₀)

a^†_−k(τ ) = α^∗_k(τ )a^†_−k(τ0) + β_k^∗(τ )ak(τ0) , (B.1) which is the so-called Bogoliubov transformation. Then standard commutation relations lead to the constraint αk(τ ) and βk(τ ) are always subject to:

|α^k(τ )|² − |β^k(τ )|² = 1 . (B.2) Then we can parametrize them in terms of the hyperbolic functions as

αk(τ ) = e^−iθkcosh rk,

βk(τ ) = e^i(θk+2ϕk)sinh rk. (B.3)